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Two-dimensional quantum well In three dimensions, the Shr ödinger Equation is:

Two-dimensional quantum well In three dimensions, the Shr ödinger Equation is:. The 3D potential well is an important system of many applications – for instance, in the modern approach to the theory of gases, such a well is the “starting point” for all considerations.

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Two-dimensional quantum well In three dimensions, the Shr ödinger Equation is:

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  1. Two-dimensional quantum well In three dimensions, the Shrödinger Equation is: The 3D potential well is an important system of many applications – for instance, in the modern approach to the theory of gases, such a well is the “starting point” for all considerations. But we will talk now only about the 2D potential well. y L0 0 x Lo

  2. Along the walls marked red the wave function must be zero: Along the lines marked blue the wave function also must be zero, which requires:

  3. Having found the wavefunction, we can write the probability density: So that the probability of finding the particle within a dxdy surface element in the well interior is: Now, energy! We have obtained before: And:

  4. So, we obtain the following expressions for the energies of the allowed quantum states in the 2D well: Let’s now examine the possible energy values:

  5. Applying the commonly used plotting scheme, we can display the possible E values using ho- rizontal “bars”: For different pairs of the nx, ny quantum numbers, one can plot the probability density function profi- les in the LxL area:

  6. Now, we want to introduce an extremelyimportant notion in Quantum Mechanics – namely, the notion of DEGENERACY Its meaning is not exactly the same as of “degeneracy” in common life situations:

  7. What does “degeneracy” mean in Quantum Mechanic? Well, if there are two or more different quantum states that happen to have exactly the same energy, we call such states “degenerate”. In the 2D quantum well most states are!

  8. We say: “The 50E energy level has a threefold degeneracy”

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